Statistics Papers
Document Type
Journal Article
Date of this Version
1986
Publication Source
The Annals of Probability
Volume
14
Issue
3
Start Page
877
Last Page
890
DOI
10.1214/aop/1176992444
Abstract
Let F and G be two continuous distribution functions that cross at a finite number of points − ∞ ≤ t1 < ⋯ < tk ≤ ∞. We study the limiting behavior of the number of times the empirical distribution function Gn crosses F and the number of times Gn crosses Fn. It is shown that these variables can be represented, as n → ∞, as the sum of k independent geometric random variables whose distributions depend on F and G only through F′(ti)/G′(ti), i = 1, …, k. The technique involves approximating Fn(t) and Gn(t) locally by Poisson processes and using renewal-theoretic arguments. The implication of the results to an algorithm for determining stochastic dominance in finance is discussed.
Keywords
asymptotic distribution, boundary crossing probability, geometric distribution, Poisson process, renewal theory, stochastic dominance algorithm, Weiner-Hopf technique
Recommended Citation
Nair, V. N., Shepp, L. A., & Klass, M. J. (1986). On the Number of Crossings of Empirical Distribution Functions. The Annals of Probability, 14 (3), 877-890. http://dx.doi.org/10.1214/aop/1176992444
Date Posted: 27 November 2017
This document has been peer reviewed.